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about midpoints and centroids

Source: RMO District 2005, 9th Grade, Problem 3

March 5, 2005
geometrygeometric transformationhomothetygeometry proposed

Problem Statement

Let ABCABC be a non-right-angled triangle and let HH be its orthocenter. Let M1,M2,M3M_1,M_2,M_3 be the midpoints of the sides BCBC, CACA, ABAB respectively. Let A1A_1, B1B_1, C1C_1 be the symmetrical points of HH with respect to M1M_1, M2M_2 and M3M_3 respectively, and let A2A_2, B2B_2, C2C_2 be the orthocenters of the triangles BA1CBA_1C, CB1ACB_1A and AC1BAC_1B respectively. Prove that: a) triangles ABCABC and A2B2C2A_2B_2C_2 have the same centroid; b) the centroids of the triangles AA1A2AA_1A_2, BB1B2BB_1B_2, CC1C2CC_1C_2 form a triangle similar with ABCABC.
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